Prandtl Lifting Line Theory

15.7.2.1 Vortex Sheet Characteristics

The conditions < u >=< w >= 0 on the vortex sheet behind a finite wing result from

• the continuity of pressure across the vortex sheet (a fluid surface),

• the tangency condition along the surface, respectively.

The existence of induced drag for a wing in incompressible, inviscid flow results from the non-zero components of v and w near the vortex sheet in the Trefftz plane, far downstream, where u = 0 (the pressure is back to the undisturbed value). The kinetic energy associated with v and w is not recoverable. There is more kinetic energy out of the control volume than into the control volume as shown by the application of the Steady Energy Equation. This excess kinetic energy comes from the work of the thrust that balances the drag.

15.7.2.2 Circulation Representation

The circulation is represented by a Fourier series

■ Г[y(t)] = 2Ub Z“=1 An sin nt
y(t) = -§ cos t, 0 < t < n

The first three modes with unit coefficients A1 = A2 = A3 = 1 are shown in Fig. 15.29.

15.7.2.3 Efficiency Factor

As seen in class, the definition of the Efficiency Factor e in terms of the Fourier coefficients is

1 – = 1 + 2 e

+——- +n

Given that the wing loading is symmetrical, only the odd modes are present. Since the “simplest” wing is assumed for the AMAT09 wing, only modes 1 and 3 are non-zero

1.053, ^ = 0.132

At

Furthermore, the mode 1 amplitude is related to the lift coefficient by

Cl = nAR Ai, ^ Ai =

The results for the three phases of flight are

• take-off: a = 18°, Cl = 2.5, ^ A1 = 0.141 , ^ A3 = 0.019

• top speed: a = -3°, Cl = 1.0 , ^ A1 = 0.056 , ^ A3 = 0.007

• power-off descent: a = 4°, Cl = 1.6 , ^ A1 = 0.09 , ^ A3 = 0.012